Depends on how the field moves relative to the iron. If it expands then contracts, which seems highly likely, the overall effect on magnetizing the iron could be surprisingly low, as far as I am aware. Though I could be wrong on this.
edit - ahh, reading through you mean during the experiment, rather than it being left magnetized afterward.
During the test it should focus field lines within it. The overall strength of the field will remain the same, but field lines will be concentrated on the cage, meaning it does not propagate as far.
Though there should be people on here that will be able to explain this better than I can. And correct me if I am talking rubbish. I am solidly an amateur on this, and there are definitely some professionals floating about.
edit - ahh, reading through you mean during the experiment, rather than it being left magnetized afterward.
During the test it should focus field lines within it. The overall strength of the field will remain the same, but field lines will be concentrated on the cage, meaning it does not propagate as far.
Though there should be people on here that will be able to explain this better than I can. And correct me if I am talking rubbish. I am solidly an amateur on this, and there are definitely some professionals floating about.